Startseite Exotic Steiner chains in Miquelian Möbius planes of odd order
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Exotic Steiner chains in Miquelian Möbius planes of odd order

  • Norbert Hungerbühler EMAIL logo und Gideon Villiger
Veröffentlicht/Copyright: 13. April 2021
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Abstract

In the Euclidean plane, two circles that intersect or are tangent clearly do not carry a finite Steiner chain of circles. We show that such exotic Steiner chains exist in finite Miquelian Möbius planes of odd order. We obtain explicit conditions in terms of the order of the plane and the capacitance of the two carrier circles for the existence, length, and number of Steiner chains.

MSC 2010: 05B25; 51E30; 51B10

Acknowledgements

We would like to thank the referee for his or her careful reading and the valuable remarks which greatly helped to improve this article.

  1. Communicated by: G. Korchmáros

References

[1] O. D. Byer, D. L. Smeltzer, A 3-D analog of Steiner’s Porism. Math. Mag. 87 (2014), 95–99. MR3193739 Zbl 1298.5101610.4169/math.mag.87.2.95Suche in Google Scholar

[2] J. L. Coolidge, A treatise on the circle and the sphere. Chelsea Publishing Co., Bronx, N.Y. 1971. MR0389515 Zbl 0251.50002Suche in Google Scholar

[3] H. S. M. Coxeter, Introduction to geometry. Wiley-Interscience 1989. MR990644 Zbl 0181.48101Suche in Google Scholar

[4] P. Dembowski, Finite geometries. Springer 1997. MR1434062 Zbl 0865.51004Suche in Google Scholar

[5] N. Hungerbühler, K. Kusejko, Steiner’s porism in finite Miquelian Möbius planes. Adv. Geom. 18 (2018), 55–68. MR3750254 Zbl 1383.0503910.1515/advgeom-2017-0027Suche in Google Scholar

[6] R. Lidl, H. Niederreiter, Introduction to finite fields and their applications. Cambridge Univ. Press 1986. MR860948 Zbl 0629.12016Suche in Google Scholar

[7] D. Pedoe, Geometry. Dover Publications, Inc., New York 1988. MR1017034 Zbl 0716.51002Suche in Google Scholar

[8] G. Villiger, A variation of Steiner’s Porism in Miquelian Möbius Planes of odd order. Master thesis, Institute of Mathematics, University of Zürich, 2018.Suche in Google Scholar

[9] P. Yiu, Rational Steiner porism. Forum Geom. 11 (2011), 237–249. MR2877262 Zbl 1287.51002Suche in Google Scholar

Received: 2018-11-13
Revised: 2019-06-22
Published Online: 2021-04-13
Published in Print: 2021-04-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 17.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/advgeom-2020-0035/html
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